Numerically solving an integral with multiple variables.

Hello. I am in dire need of help. I'm trying to numerically solve this integral, the Formula is on the third page (of the attached pdf) -- the F(1/2). I've tried different function handles but am coming up with nothing. Here are some things that I tried. I want Vg and Vd to be varying with this. Don't give me the whole answer but where can I even start? How do I start? I'm just unsure how to integrate at this point.

% INITIALIZE MATLAB
close all;
clc;
clear all;

syms x
syms Ro
syms Vd
syms Vg
syms CoRo %Constant of p
syms p
Vt = 0.7; 
Vg = 2; 

CoRo = 1.652e+16;
%Vg = 0:0.2:1.2;
% Ro
p = CoRo*(Vg-Vt);
% FUNCTIONS AND INTERVAL
a = @(Vd) 1+exp(Vd);
b = @(Vd)sqrt(a^2+4*exp(Vd)*(exp(p)-1));
u = @(Vd) log(b-(1+exp(Vd))-log(2));
f = @(u) sqrt(u)/(1+exp(u-x));

%a = inline(1+exp(Vd));
%b = inline('sqrt(a^2+4*exp(Vd)*(exp(p)-1))','Vd');
%u = inline('log(b-(1+exp(Vd))-log(2))','Vd');
%f = inline('sqrt(u)/(1+exp(u-x))','u');

d = integral(f,0,5)

% INTEGRAL
    %Starting value
    %s = 0;
    %Ending value
    %e = 5;

    %F = integral(f,s,e);

% NUMBER OF POINTS
N = 10;

% INTEGRAL OF f

Answer 1

[Torsten]

f = @(u,x)sqrt(u)./(1+exp(u-x));
F = @(x)integral(@(u)f(u,x),0,Inf,'ArrayValued',true);
F([2 4])

[Ausamah Hobbi]

Thank you! It works. Now just some questions for my own reference, if you don't mind.

• What does 'ArrayValued' do?
• What does F([2 4]) do? Is it just values for x at 2 and 4?
• What does the '.' do in the first line?

My next part of this is to vary the value of Vg. Does what I have "Vg = 0:0.2:1.2;" work? As well as varying Vd for which I have an excel file for. Any suggestions there?

Thank you very much!